集值 sub-topical 映射的抽象凸理论

Abstract convex theory for set-valued sub-topical mappings

摘要

sub-topical 函数作为一类重要的抽象凸函数，在优化理论等领域具有较大的研究价值。针对抽象凸分析中标量 sub-topical 函数无法直接处理具有序结构的集值优化的问题，本文将 sub-topical 函数的相关概念从标量情形拓展至集值情形，以处理更广泛的优化问题。借助向量空间中的弱上确界概念，首先构造了一类具有特定性质的集值基本映射作为承托元素，然后引入集值 sub-topical 映射的承托集，并在此基础上建立了其相对于该基本映射族的上包络刻画，最后为集值 sub-topical 映射构建了相应的抽象凸理论，为后续在集值优化等领域的应用奠定了基础。

Abstract

Sub-topical functions constitute a significant class of abstract convex functions and play a pivotal role in optimization theory. However, the scalar framework of sub-topical functions faces limitations when addressing set-valued optimization problems equipped with order structures. To bridge this gap, this paper generalizes the concept of sub-topicality from scalar-valued mappings to set-valued mappings. By leveraging the notion of weak supremum in vector spaces, we first construct a class of set-valued elementary mappings with specific algebraic properties to serve as supporting elements. Subsequently, we introduce the support set for these set-valued sub-topical mappings and establish their upper envelope representation with respect to this family of elementary mappings. Finally, we develop a comprehensive abstract convexity theory for set-valued sub-topical mappings. This framework lays a solid theoretical foundation for future research in set-valued optimization and related fields.

关键词

集值 sub-topical 映射 / 抽象凸分析 / 偏序上确界 / 集值承托

Key words

set-valued sub-topical mapping / abstract convex analysis / partially ordered supremum / set-valued support

作者贡献声明

高瑞敏负责论文的构思与设计、论文的撰写、内容修订与语言润色，并承担文中定理和命题的理论证明及结果刻画。姚朝李提供学术指导，对文中定义的合理性和定理证明的严谨性进行审查，对论文内容提出关键修改意见和补充实例建议，并进行逻辑调整和经费支持。

AI使用声明

本文未使用人工智能（AI）协助撰写。

利益冲突声明

本人及所有合著者郑重声明，在本文设计、实施及论文撰写过程中，不存在已知的可能影响本论文所报告工作的竞争性经济利益或个人关系。

伦理声明

不适用：本研究未涉及人类或动物的任何实验。

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